Abelian Product of Free Abelian and Free Lie Algebras

Let $F_{n}$ be a free Lie Algebra of finite rank $n$ and $A$ be a free abelian Lie algebra of finite rank $m\geq 0$ . We investigate the properties of the generating sets and subalgebras of the abelian product $A\ast _{ab}F_{n}$. Moreover these properties are used to solve the membership problem for $A\ast _{ab}F_{n}$.